Adviser ： 高永安 Student ： 林柏廷

1. Interpolation in Digital Modems-Part II: Implementation and PerformanceLars Erup, Member, IEEE, Floyd M. Gardner, Fellow, IEEE, and Robert A. Harris, Member, IEEEIEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 41, NO. 6, JUNE 1993 Adviser：高永安 Student：林柏廷

2. Outline • Introduction • Polynomial-based filter • Filter structures • Performance simulation • Closed-loop simulation • Summary

3. Introduction (I) • Part I of this work[1] presented a fundamental equation for digital interpolation of data signals： • Where ： basepoint index i： filter index ： fractional interval [1] F.M. Gardner, “Interpolation in digital modems- Part I: Fundamentals,” IEEE Trans. Commun., vol. 41, pp. 502-508, Mar. 1993.

4. Introduction (II)

5. Introduction (III) • This paper, in Section II, explores properties of simple polynomial-based filters - a category that includes the classical interpolating polynomials. • Several digital structures for performing the computations in (1) have been considered, and are discussed in Section III. • Simulations have been run on several candidate filters to evaluate their effect on modem performance, with results given in Section IV.

6. Introduction (IV) • The design of a complete timing control loop, based on the NCO method devised in [l], is illustrated in Section V.

7. Polynomial-based filter (I) • Anextensive literature exists on polynomial interpolation, which is a small but important subclass of polynomial-based filters. • They are easy to describe. • Good filter characteristics can be achieved. • A special FIR structure (to be described below) permits simple handling of filter coefficients. The structure is applicable only to polynomials.

8. Polynomial-based filter (II) • Any classical polynomial interpolation can be described in terms of its Lagrange coefficients. • The Lagrange coefficientformulas constitute the filter impulse response hI(t), sothat the latter is piecewise polynomial for classical polynomial interpolation.

9. Polynomial-based filter (III) • The time-continuous impulse response associated with a linear interpolator is simply an isosceles triangle, with hI1(0)= 1,and basewidth of 2Ts ; see Fig. 1.

10. Polynomial-based filter (IV) • Its frequency response is plotted in Fig. 2.

11. Polynomial-based filter (V) • Fig. 3 shows a signal at 2 samples/symbol with 100% excess bandwidth, together with the frequency response of the cubic interpolator.

12. Filter structures (I) • Linear interpolation, or the four-point piecewise-parabolic impulse response provide simple formulas and are well-suited for rapid direct evaluation of filter coefficients. • Linear interpolation filter can be performed by the simple formula

13. Filter structures (II) • Another approach, better suited for signal interpolation by machine, has been devised by Farrow[10]. Let the impulse response be piecewise polynomial in each Ts,-segment, i = I1 to I2 : [10]C.W. Farrow, “A continuously variable digital delay element,” in Proc. IEEE Int. Symp. Circuits & Syst., Espoo, Finland, June 6-9, 1988, pp. 2641-2645.

14. Filter structures (III) • For a cubic interpolation

15. Filter structures (VI)

16. Filter structures (V)

17. Filter structures (VI)

18. Performance simulation (I) • Fig. 7 shows, as an example, the frequency responses of a 註1 filter with and without compensation for the CUF of a linear interpolator at 2samples/symbol. 註1 = root cosine roll-off filter with 100% excess bandwidth.

19. Performance simulation (II) • Most results were obtained using semi-analytic error probability estimation, using BPSK modulation and averaging over 1023 symbols (the data source was a PN sequence generated by a 10-stage shift register).

20. Performance simulation (III)

21. Performance simulation (IV)

22. Performance simulation (V)

23. Closed-loop simulation (I) • we employed the Data-Transition Tracking Loop (DTTL) algorithm [11,12] to compute the timing error signal . For BPSK, the expression is [11] F.M. Gardner, “A BPSWQPSK timing-error detector for sampled receivers,”IEEE Trans. Commun., vol. COM-34, pp. 423429, May 1986. [12] W.C. Lindsey and M.K. Simon, Telecommunications Systems Engineering. Englewood Cliffs, NJ: Prentice-Hall, 1973, ch. 9.

24. Closed-loop simulation (II) • Where is the real part of the sampled baseband signal, is a hard-decision on , is an symbol index.

25. 101010…. Data sequence pseudorandom sequence with 50% CRO filtering Closed-loop simulation (III) • Fig. 14 shows (additive noise-free)s-curves of the DTTL algorithm for BPSK signals with unit power at the input to the receiver filter.

26. Closed-loop simulation (IV) • Consider the loop in tracking mode, with the NCO receiving a constant control value and overflowing at regular intervals 註2. • Input to the loop is a phase step of one-quarter symbol period. Fig. 15-17. • Response to a step in symbol frequency of 0.1%. Fig 18-20. 註2The NCO in [1] had modulus 1 whereas the one considered here has modulus Ti/Ts.

27. Closed-loop simulation (V)

28. Closed-loop simulation (VI) At 4 samples per symbol, 0.1% of the symbol rate corresponds to a shift of 1 sample in 250 symbols. This is confirmed by Fig. 20.

29. Summary (I) • The Farrow structure [l0] appears to be particularly interesting for high-speed operation as it only requires the transfer of a single control value and uses a small number of multiplications. • For more critical applications, four-point interpolation using piecewise-parabolic interpolating filters (α = 0.5) gives excellent performance with only moderate complexity increase. [10]C.W. Farrow, “A continuously variable digital delay element,” in Proc. IEEE Int. Symp. Circuits & Syst., Espoo, Finland, June 6-9, 1988, pp. 2641-2645.

30. Summary (II) • Closed-loop simulations have shown that the NCO-based control system has many similarities to a conventional analogue phase-locked loop.